cp's OEIS Frontend

A332282 Numbers whose unsorted prime signature is not unimodal.

%I A332282 #5 Feb 16 2025 08:33:59
%S A332282 300,588,600,980,1176,1200,1452,1500,1960,2028,2100,2205,2352,2400,
%T A332282 2420,2904,2940,3000,3300,3380,3388,3468,3900,3920,4056,4116,4200,
%U A332282 4332,4410,4704,4732,4800,4840,5100,5445,5700,5780,5808,5880,6000,6348,6468,6600,6615
%N A332282 Numbers whose unsorted prime signature is not unimodal.
%C A332282 The unsorted prime signature of a positive integer (row n of A124010) is the sequence of exponents it is prime factorization.
%C A332282 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A332282 Also Heinz numbers of integer partitions with non-unimodal run-lengths. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A332282 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e A332282 The sequence of terms together with their prime indices begins:
%e A332282    300: {1,1,2,3,3}
%e A332282    588: {1,1,2,4,4}
%e A332282    600: {1,1,1,2,3,3}
%e A332282    980: {1,1,3,4,4}
%e A332282   1176: {1,1,1,2,4,4}
%e A332282   1200: {1,1,1,1,2,3,3}
%e A332282   1452: {1,1,2,5,5}
%e A332282   1500: {1,1,2,3,3,3}
%e A332282   1960: {1,1,1,3,4,4}
%e A332282   2028: {1,1,2,6,6}
%e A332282   2100: {1,1,2,3,3,4}
%e A332282   2205: {2,2,3,4,4}
%e A332282   2352: {1,1,1,1,2,4,4}
%e A332282   2400: {1,1,1,1,1,2,3,3}
%e A332282   2420: {1,1,3,5,5}
%e A332282   2904: {1,1,1,2,5,5}
%e A332282   2940: {1,1,2,3,4,4}
%e A332282   3000: {1,1,1,2,3,3,3}
%e A332282   3300: {1,1,2,3,3,5}
%e A332282   3380: {1,1,3,6,6}
%t A332282 unimodQ[q_]:=Or[Length[q]==1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332282 Select[Range[1000],!unimodQ[Last/@FactorInteger[#]]&]
%Y A332282 The opposite version is A332642.
%Y A332282 These are the Heinz numbers of the partitions counted by A332281.
%Y A332282 Non-unimodal permutations are A059204.
%Y A332282 Non-unimodal compositions are A115981.
%Y A332282 Non-unimodal normal sequences are A328509.
%Y A332282 Cf. A001523, A007052, A056239, A112798, A124010, A227038, A332280, A332284, A332286, A332639, A332643, A332671.
%K A332282 nonn
%O A332282 1,1
%A A332282 _Gus Wiseman_, Feb 19 2020